I gave the following question on my last physical science (physical science for non-science majors) test:

Block A has a mass of 10 kg and volume of 100 cm3. Block B has a mass of 1 kg and a volume of 1 cm3. Which block has the greater density? Which block has the greater volume?

The results of this question are interesting. Of the respondents, 33 answered both parts in a meaningful way. 8 of these 33 answered that the one with the greatest density had the greatest volume (that is 24%). Most of these 24% said that 1 cm3 was greater than 100 cm3. However, there were some that said the density of the 1 cm3 block had a smaller density because the volume was smaller.

Can around 20% of the class really think that 1 is greater than 100? I don’t think so. Rather, they are falling into the classic trap of thinking that says:

If two things are compared, the one with the greater attribute will be greater in everything.

You can see this with density, mass and volume quite easily. Which has a greater mass, a tiny rock or a huge pile of pillows? Some students and people will say the rock. Really, they will. Here is another: A bug collides with a bus. Which exerts a greater force on the other? Well, the bus is bigger so it should exert a greater force. You get the idea.

I think this is similar, but different than the confusion between a quantity and the rate that quantity changes (like velocity and acceleration).

Comments

Maybe it would have been less confusing if the masses were the same. Or maybe they just can’t picture “volume” in their heads and would have answered more correctly if the question had more imagery, as in, “Block A is 10 cm long, 10 cm wide, and 1 cm high” and “Block B is 1 cm long, 1 cm wide, and 1 cm high.”

In both cases they are using their intuitive physics ideas that are somewhat flawed. These are usually called preconceptions or misconceptions in physics education research. The thing is with these kind of questions people tend to (unintentionally) disregard their knowledge of actual physical laws and models and just stick with their intuition.

I think the confusion arises because you mixed together intensive (density) and extensive (volume) quantities. You could probably create a similar confusion by asking a question about the amount heat energy in two metal bars of different sizes compared to their relative temperatures. Probably many students would say that the bar containing greater heat energy was ‘hotter’ than the other, regardless of their actual temperatures.

Some of the students no doubt had the conceptual problem you describe. However, with no basis in fact, I will guess that a majority of the minority that missed this question just forgot or did not know the definitions of volume, mass and density.

I can’t for the life of me imagine where college students would have formed the ideas in their 18 years of existence necessary to believe that “If two things are compared, the one with the greater attribute will be greater in EVERYTHING.” What thing have these 18 year olds ever encountered which, being greater in one attribute, dictates that it is greater in all attributes?

But you are right, of course, we should address the problems that can be addressed.

Clearly, some that missed this just weren’t paying attention. Also, I doubt any student thinks that greater always means greater. The problem is probably that they are looking at this at a very superficial level (because that is the way they are used to taking tests).

You have to use–gasp–math to figure out which block has the higher density. But the volumes are given. Why was the percentage of people who attempted this question and got it right not 100 percent?

As for relative masses, here’s a trick question: Which weighs more, a pound of feathers or a pound of gold? In this case, the pound of feathers is heavier, because English weight measures for gold are not the same as for feathers. Gold is measured in troy weight, where 12 ounces make up a pound. Feathers are measured in avoirdupois weight (as are most things other than precious metals), where the ounce is smaller, but not by enough to make up for the fact that there are 16 of those ounces in a pound. According to Wikipedia, the troy pound is about 373 g, versus 454 g for the avoirdupois pound, under normal Earth gravity.

#10 proves its better to be metric, a kilo is a kilo, the question has the proper answer in that system. Technically of course you are to put the question in terms of slugs, because you would have to state sitting next to each other in addition for the pound case. In fact by choice of location you could make the answer to 10 be any answer you like.
(I know picky picky but weight is a poor way to as the question)